Theoretical limits on verifying self-improving systems without risking unbounded utility.
problem Formalizing and proving the limits of safety verification for self-improving systems.
method Developed dual conditions and used Holder's inequality, NP counting method, and Lipschitz bounds to establish impossibility and ceiling results.
result A classifier-based safety gate cannot simultaneously permit unbounded beneficial self-modification and bounded cumulative risk.
The cellular tree classifier model addresses a fundamental problem in the design of classifiers for a parallel or distributed computing world: Given a data set, is it sufficient to apply a majority rule for classification, or shall one split the data into two or more parts and send each part to a potentially different …
New research shows fair data representations are impossible for different tasks.
problem Achieving fairness in machine learning models trained on various tasks.
method Analyzing the limits of fair data representations.
result No representation can guarantee fairness for different tasks trained on it.
Consider a classification problem where we have both labeled and unlabeled data available. We show that for linear classifiers defined by convex margin-based surrogate losses that are decreasing, it is impossible to construct any semi-supervised approach that is able to guarantee an improvement over the supervised clas…
Paper investigates learnability of OOD detection under various conditions.
problem Learnability of OOD detection under different scenarios.
method Investigates PAC learning theory, proves impossibility theorems, and provides necessary and sufficient conditions.
result Some conditions for learnability of OOD detection may not hold in practical scenarios.
Classifier-based AI safety gates fail in self-improvement, even with advanced verification methods.
problem Maintaining reliable oversight of AI systems as they improve over iterations.
method Comprehensive empirical testing on neural controllers and MuJoCo benchmarks, using various classifiers and verification methods.
result Classifier-based safety gates fail in maintaining reliable oversight, even with advanced verification methods.
Paper investigates OOD detection learnability under various conditions.
problem Learnability of OOD detection under diverse and unknown test data.
method PAC learning theory applied to OOD detection, proving impossibility theorems and necessary conditions.
result Some conditions for learnability hold in practical scenarios.
The majority of traditional classification ru les minimizing the expected probability of error (0-1 loss) are inappropriate if the class probability distributions are ill-defined or impossible to estimate. We argue that in such cases class domains should be used instead of class distributions or densities to construct …
New method uses path signatures for efficient likelihood estimation in time-series data.
problem Intractable likelihood functions in complex dynamic models.
method Kernel classifier based on path signatures for sequential data.
result Path signatures yield highly performant classifiers, even with low sample numbers.
AdaBoost's success explained through noise influence measure.
problem Understanding why AdaBoost is a successful classifier.
method Introduced a measure of noise influence (ION) to explain AdaBoost's success.
result ION decreases with iteration number and base learner complexity.
Empirical risk minimization (ERM), with proper loss function and regularization, is the common practice of supervised classification. In this paper, we study training arbitrary (from linear to deep) binary classifier from only unlabeled (U) data by ERM. We prove that it is impossible to estimate the risk of an arbitrar…
Improved conformal prediction for better conditional coverage of classifier predictions.
problem Achieving exact conditional coverage in finite samples for prediction sets.
method Developed a variant of conformal prediction targeting coverage conditional on confidence and trust score.
result Empirically improved conditional coverage properties compared to standard conformal prediction.
Classifies nilpotent Lie groups with specific G2-structures.
problem Identifying 7D nilpotent Lie groups with purely coclosed G2-structures. method Examined all 7D nilpotent Lie algebras, classified them, and verified the existence or non-existence of G2-structures. result Provided a complete classification of 7D nilpotent Lie groups with purely coclosed G2-structures. Four-dimensional Einstein Dehn filling is impossible.
problem Complex-hyperbolic Einstein Dehn filling in four dimensions.
method Proof of impossibility.
result Complex-hyperbolic Einstein Dehn filling cannot be performed in dimension four.
Neural networks approximate likelihood ratios for complex models.
problem Difficulty in computing likelihood ratios for modern models.
method Applying the likelihood ratio trick with neural network classifiers.
result Different neural network setups can approximate likelihood ratios with varying performance.
New approach to certifiably robust neural networks using Boolean function perspective.
problem Lack of principled understanding and certified robustness for ℓ∞ perturbations. method New perspective on Boolean functions, deriving impossibility results, and developing a unified Lipschitz network.
result Unified Lipschitz network that bypasses expressive power limitations and achieves better certified robustness.
Universally valid ground truth is almost impossible to obtain or would come at a very high cost. For supervised learning without universally valid ground truth, a recommended approach is applying crowdsourcing: Gathering a large data set annotated by multiple individuals of varying possibly expertise levels and inferri…
Machine fairness is impossible to achieve fully due to historical biases.
problem Machine learning models inherit biases from historical data, making it impossible to satisfy fairness metrics simultaneously.
method Presented a causal perspective to the impossibility theorem of fairness.
result It is impossible to satisfy fairness metrics like demographic parity, equal opportunity, and equalized odds simultaneously.
Vapnik-Chervonenkis (VC) dimension is a fundamental measure of the generalization capacity of learning algorithms. However, apart from a few special cases, it is hard or impossible to calculate analytically. Vapnik et al. [10] proposed a technique for estimating the VC dimension empirically. While their approach behave…
In this article we prove the impossibility of some disentanglement puzzles, first building mathematical models that reflect the essential characteristics of these puzzles.
In various approaches to learning, notably in domain adaptation, active learning, learning under covariate shift, semi-supervised learning, learning with concept drift, and the like, one often wants to compare a baseline classifier to one or more advanced (or at least different) strategies. In this chapter, we basicall…
TWM doesn't reduce delta in PDLPs, proving impossibility.
problem TWM in PDLPs doesn't uniformly reduce portfolio delta.
method Proved TWM's condition is self-contradictory and showed impossibility.
result No TWM can uniformly reduce portfolio delta.
An impossibility result shows limitations in learning symmetries and equivariant functions.
problem Learning symmetries and equivariant functions simultaneously is impossible under certain conditions.
method Careful study of approximation for groups and semigroups, analysis of neural networks.
result Linearly equivariant networks can be used to learn equivariant functions, but group-convolutional networks have limitations.
Lower bounds show learning mixtures of linear classifiers is nearly impossible.
problem Learning mixtures of linear classifiers under Gaussian covariates.
method Statistical Query (SQ) lower bounds and new spherical designs.
result Complexity of any SQ algorithm is \( n^{\mathrm{poly}(1/Δ) \log(r)} \), where Δ is the pairwise \(\ell_2\)-separation.
Large language models can't efficiently reason conditionally in a distribution-free setting.
problem Impossibility of conditional PAC-efficient reasoning in large language models.
method Proof of impossibility in a distribution-free setting for non-atomic input spaces.
result Any algorithm achieving conditional PAC efficiency must defer to the expert model with high probability.
No feature ranking can be faithful, stable, and complete when features are collinear.
problem The impossibility of creating a feature ranking that is simultaneously faithful, stable, and complete when features are collinear.
method Proving the impossibility, quantifying it for four model classes, resolving it via ensemble averaging (DASH), and machine-verifying it with Lean 4 theorems.
result No method lies outside the dichotomy of faithful-complete methods (unstable, with rankings that flip up to 50% of the time) and ensemble methods (stable, reporting ties for symmetric features).
Two impossibility theorems show formal alignment certification is impossible for AI systems.
problem Formal certification of AI alignment over open-ended domains is impossible.
method Two independent impossibility theorems: Semantic and Statistical barriers.
result No procedure can simultaneously satisfy soundness, completeness, and tractability.
Paper proves impossibility of three desirable properties in node embedding.
problem Understanding limitations of node embedding methods.
method Axiomatic approach to node embedding, proving impossibility of three properties.
result No node embedding method can satisfy all three desirable properties simultaneously.
Anchors explains text classifiers by highlighting key words.
problem Interpreting machine learning models, especially for text classifiers.
method Formalizes Anchors' algorithm and analyzes its behavior on linear text classifiers.
result Anchors produces meaningful results on linear text classifiers.
A new method for variable importance measures without impossible data.
problem Using impossible data for variable importance measures in black box models.
method Cohort Shapley, a method grounded in economic game theory using only observed data.
result Cohort Shapley provides a more trustworthy explanation of black box models' decisions.
We continue the study of statistical/computational tradeoffs in learning robust classifiers, following the recent work of Bubeck, Lee, Price and Razenshteyn who showed examples of classification tasks where (a) an efficient robust classifier exists, in the small-perturbation regime; (b) a non-robust classifier can be l…
The study of tiling homology on flat surfaces, proving impossibility of certain tilings.
problem Proving the non-existence of polyomino tilings on specific square-tiled surfaces.
method Study of homology groups for topological tilings, using coloring proofs.
result Several results about the non-existence of polyomino tilings on certain square-tiled surfaces.
Researchers prove it's impossible to partially recover graph alignments in certain conditions.
problem Recovering vertex correspondence between two random graphs with correlated edges.
method Used the probabilistic method to build automorphisms between tree components of a subcritical Erdös-Rényi graph.
result Proved an impossibility result for partial recovery in the sparse regime with constant average degree and correlation.
Ensemble models refer to methods that combine a typically large number of classifiers into a compound prediction. The output of an ensemble method is the result of fitting a base-learning algorithm to a given data set, and obtaining diverse answers by reweighting the observations or by resampling them using a given pro…
New method calibrates uncertainty estimates for image classifiers without labeled data.
problem Uncertainty estimates for modern classifiers are unreliable without labeled calibration data.
method Calibrates uncertainty estimates using unlabeled examples for distribution shifts.
result Proposes a method that provides excellent uncertainty estimates under natural distribution shifts.
Unified approach optimizes neural network training for various metrics.
problem Training and evaluation of neural network binary classifiers often use different metrics.
method Combines differentiable approximation and probabilistic soft sets.
result Effective in optimizing for metrics like F1-Score across various domains.
Hass and Scott's example of a 4-valent graph on the 3-punctured sphere that cannot be realized by geodesics in any metric of negative curvature is generalized to impossible configurations filling surfaces of genus n with k punctures for any n and k.
Identification of disease subtypes and corresponding biomarkers can substantially improve clinical diagnosis and treatment selection. Discovering these subtypes in noisy, high dimensional biomedical data is often impossible for humans and challenging for machines. We introduce a new approach to facilitate the discovery…
Numbers and numerical vectors account for a large portion of data. However, recently the amount of string data generated has increased dramatically. Consequently, classifying string data is a common problem in many fields. The most widely used approach to this problem is to convert strings into numerical vectors using …
New protocol identifies impossible edge orientations in causal graphs.
problem Causal-discovery algorithms cannot distinguish edge directions without assumptions.
method Discrete impossibility certificates and oracle queries.
result Upper bound of 1+K expert interactions for DAG recovery. A popular methodology for building binary decision-making classifiers in the presence of imperfect information is to first construct a non-binary "scoring" classifier that is calibrated over all protected groups, and then to post-process this score to obtain a binary decision. We study the feasibility of achieving vari…
In biospectroscopy, suitably annotated and statistically independent samples (e. g. patients, batches, etc.) for classifier training and testing are scarce and costly. Learning curves show the model performance as function of the training sample size and can help to determine the sample size needed to train good classi…
Researchers prove inner product recovery is impossible in latent space models.
problem Recovering inner products in latent space models with random geometric graphs.
method Rate-distortion theory applied to Gaussian or spherical latent locations.
result Impossible to recover inner products if dimensionality exceeds nh(p), matching positive results' conditions. We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspensio…
Since time immemorial, people have been looking for ways to organize scientific knowledge into some systems to facilitate search and discovery of new ideas. The problem was partially solved in the pre-Internet era using library classifications, but nowadays it is nearly impossible to classify all scientific and popular…
Paper proves impossible for large language models to control hallucinations without sacrificing other properties.
problem Achieving truthful knowledge representation, semantic information conservation, and knowledge-constrained optimality simultaneously in large language models.
method Modeling inference as an auction of ideas, using mechanism design, proper scoring rules, and transformer architecture analysis.
result No LLM can simultaneously achieve all four essential properties without violating at least one.
In binary classification framework, we are interested in making cost sensitive label predictions in the presence of uniform/symmetric label noise. We first observe that 0-1 Bayes classifiers are not (uniform) noise robust in cost sensitive setting. To circumvent this impossibility result, we present two schemes; un…
We solve the multi-criteria benchmarking problem by formalizing it as a social choice problem and identifying conditions for meaningful rankings.
problem Aggregating multiple metrics into a single ranking for models in benchmarking problems.
method Formalizing multi-criteria benchmarking as a social choice problem and identifying sufficient conditions for meaningful rankings.
result We prove that meaningful multi-criteria benchmarking becomes possible under certain preference conditions (single-peaked, group-separable, distance-restricted).